PepsiCo Inc. has agreed to buy one of its fiercest critics, seltzer-machine maker SodaStream International Ltd., for $3.2 billion, the latest move by the beverage-and-food giant to broaden its offerings beyond sugary sodas and salty snacks.

Israel-based SodaStream makes countertop machines that allow consumers to carbonate tap water at home by filling a reusable bottle and adding flavors if desired. As consumers have shifted from sugary soft drinks to bottled water, the Nasdaq-listed company has pivoted to promote itself...

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